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# PhaseEstimation¶

class PhaseEstimation(num_evaluation_qubits, quantum_instance=None, sampler=None)[source]

Bases: qiskit.algorithms.phase_estimators.phase_estimator.PhaseEstimator

Run the Quantum Phase Estimation (QPE) algorithm.

This runs QPE with a multi-qubit register for reading the phases  of input states.

The algorithm takes as input a unitary $$U$$ and a state $$|\psi\rangle$$, which may be written

$|\psi\rangle = \sum_j c_j |\phi_j\rangle,$

where $$|\phi_j\rangle$$ are eigenstates of $$U$$. We prepare the quantum register in the state $$|\psi\rangle$$ then apply $$U$$ leaving the register in the state

$U|\psi\rangle = \sum_j \exp(i \phi_j) c_j |\phi_j\rangle.$

In the ideal case, one then measures the phase $$\phi_j$$ with probability $$|c_j|^2$$. In practice, many (or all) of the bit strings may be measured due to noise and the possibility that $$\phi_j$$ may not be representable exactly by the output register. In the latter case the probability for each eigenphase will be spread across bitstrings, with amplitudes that decrease with distance from the bitstring most closely approximating the eigenphase.

The main input to the constructor is the number of qubits in the phase-reading register. For phase estimation, there are two methods:

first. estimate, which takes a state preparation circuit to prepare an input state, and

a unitary that will act on the input state. In this case, an instance of qiskit.circuit.PhaseEstimation, a QPE circuit, containing the state preparation and input unitary will be constructed.

second. estimate_from_pe_circuit, which takes a quantum-phase-estimation circuit in which

the unitary and state preparation are already embedded.

In both estimation methods, the QPE circuit is run on a backend and the frequencies or counts of the phases represented by bitstrings are recorded. The results are returned as an instance of PhaseEstimationResult.

Reference:

: Michael A. Nielsen and Isaac L. Chuang. 2011.

Quantum Computation and Quantum Information: 10th Anniversary Edition (10th ed.). Cambridge University Press, New York, NY, USA.

Parameters
• num_evaluation_qubits (int) – The number of qubits used in estimating the phase. The phase will be estimated as a binary string with this many bits.

• quantum_instance (QuantumInstance | Backend | None) – Pending deprecation: The quantum instance on which the circuit will be run.

• sampler (BaseSampler | None) – The sampler primitive on which the circuit will be sampled.

Raises

AlgorithmError – If neither sampler nor quantum instance is provided.

Methods

 construct_circuit Return the circuit to be executed to estimate phases. estimate Build a phase estimation circuit and run the corresponding algorithm. estimate_from_pe_circuit Run the phase estimation algorithm on a phase estimation circuit